In (4.23.1) and (4.23.2) the integration paths may
not pass through either of the points t=±1. The function
(1-t2)1/2 assumes its principal value when t∈(-1,1); elsewhere on
the integration paths the branch is determined by continuity. In
(4.23.3) the integration path may not intersect ±i. Each of
the six functions is a multivalued function of z. Arctan⁡z and
Arccot⁡z have branch points at z=±i; the other four functions
have branch points at z=±1.

The principal values (or principal branches) of the inverse sine,
cosine, and tangent are obtained by introducing cuts in the z-plane as
indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the
integration paths in (4.23.1)–(4.23.3) not to cross
these cuts. Compare the principal value of the logarithm
(§4.2(i)). The principal branches are denoted by arcsin⁡z,
arccos⁡z, arctan⁡z, respectively.
Each is two-valued on the corresponding cuts,
and each is real on the part of the real
axis that remains after deleting the intersections with the corresponding cuts.

The principal values of the inverse cosecant, secant, and cotangent are given
by

§4.23(iv) Logarithmic Forms

Notes:

To verify (4.23.19), denote the right-hand side by ϕ⁡(z),
and the domain ℂ\(-∞,-1]∪[1,∞) by D.
If z=x∈(-1,1), then ϕ′⁡(x)=(1-x2)-1/2 and
ϕ⁡(0)=0. Hence (4.23.19) applies; compare
(4.23.1) with Arcsin replaced by arcsin. We may now
extend (4.23.19) to the rest of D simply by showing that
ϕ⁡(z) is analytic on D; compare §1.10(ii). Since the
principal value of (1-z2)1/2 is analytic on D, the only
possible singularities of ϕ⁡(z) occur on the branch cut of the
logarithm, that is, when (1-z2)1/2=-i⁢z-t with
t∈[0,∞). By squaring the last equation we see that
(1-z2)1/2+i⁢z is real only when z lies on the imaginary axis,
and it is then positive.
The proofs of (4.23.22), (4.23.23),
(4.23.26) are similar, or in the case of
(4.23.22) we may simply refer to (4.23.16).

Originally the factor sign⁡(y) was missing from the second term on the right side
of this equation. Also, the originally stated condition x∈[-1,1] for this equation,
stated on the line following (4.23.36), was replaced with the more general
condition ±z∉(1,∞).

Originally the factor sign⁡(y) was missing from the second term on the right side
of this equation. Also, the originally stated condition x∈[-1,1] for this equation,
stated on the line following (4.23.36), was replaced with the more general
condition ±z∉(1,∞).